Abstract

We propose the weighted expected sample size (WESS) to evaluate the overall performance on the indifference-zones for three composite hypotheses’ testing problem. Based on minimizing the WESS to control the expected sample sizes, a new sequential test is developed by utilizing two double sequential weighted probability ratio tests (2-SWPRTs) simultaneously. It is proven that the proposed test has a finite stopping time and is asymptotically optimal in the sense of asymptotically minimizing not only the expected sample size but also any positive moment of the stopping time on the indifference-zones under some mild conditions. Simulation studies illustrate that the proposed test has the smallest WESS and relative mean index (RMI) compared with Sobel-Wald and Whitehead-Brunier tests.

1. Introduction

Let be independent and identically distributed (i.i.d.) random variables whose common density function (with respect to some nondegenerate measure ) belongs to the exponential family where is a convex function and is the natural parameter space with . The problem of interest is the following three composite hypotheses’ testing problem: where . For example, in clinical trial applications, in order to compare the effects of two drugs (Goeman et al. [1]), the equivalence trial versus would be more realistically stated as (inferiority), (equivalence), and (superiority), where is the difference of effect between two drugs. The sequential testing of three or more hypotheses has been applied to a variety of engineering problems such as pattern recognition (Fu [2]; McMillen and Holmes [3]), multiple-resolution radar detection (Bussgang [4]), products comparisons (Anderson [5]), and others (Li et al. [6]). The intervals of and are usually called indifference-zones and denoted by .

Published work on this problem has taken two main approaches. Pavlov [7], Baum and Veeravalli [8], and Dragalin et al. [9, 10] studied the class of tests motivated by the Bayesian framework. The second approach has focused on extending the sequential probability ratio test (SPRT) and double sequential probability ratio test (2-SPRT) to incorporate more than two hypotheses, such as Sobel and Wald [11], Armitage [12], Simons [13], Lorden [14], Whitehead and Brunier [15], and Li and Pu [16, 17]. Dragalin and Novikov [18] studied the problem of testing several composite hypotheses with an indifference-zone for an unknown parameter. Lai [19] considered the multihypothesis testing problem where some or all of these hypotheses are composite.

Among others, the tests proposed by Sobel and Wald [11] and Whitehead and Brunier [15] are usually used in practice for problem (2). Specifically, Sobel and Wald [11] proposed carrying out simultaneous SPRTs of versus and versus . However, when the true parameter is in the indifference-zones, the expected sample size of the Sobel-Wald test can be considerably larger than that of a fixed-sample-size test plan. Moreover, it is untruncated such that the number of observations required can not be predetermined, an undesirable property in many practical situations such as medical trial. To reduce the maximum expected sample size, Whitehead and Brunier [15] applied two 2-SPRTs instead of two SPRTs for the component tests, at the cost of larger expected sample sizes when the true parameter does not belong to the indifference-zones.

For one-sided composite hypotheses, in order to control the expected sample sizes, Wang et al. [20] proposed the double sequential weighted probability ratio test (2-SWPRT) based on mixture likelihood ratio statistics and showed that the 2-SWPRT is an asymptotically overall optimal test in the sense of asymptotically minimizing the expected sample sizes on the indifference-zone. Motivated by the attractive properties of the 2-SWPRT, we extend the existing work on problem (2) from pointwise optimality to overall performance optimality when there are different concerns of interest on different s. In particular, we propose an optimality criterion to evaluate the overall performance of sequential test plans on the indifference-zones for three composite hypotheses and correspondingly develop a new sequential test for problem (2) by utilizing two 2-SWPRTs as the component tests to reduce the expected sample sizes. We show the proposed test has a finite stopping time and is asymptotically optimal in the sense of asymptotically minimizing not only the expected sample size but also any positive moment of the stopping time on the indifference-zones. Simulation studies show that the proposed test not only has the smallest WESS compared with Sobel-Wald and Whitehead-Brunier tests, but also is superior to the Whitehead-Brunier test and comparable with the Sobel-Wald test when the true parameter does not belong to the indifference-zones. Moreover, the RMI also shows the proposed test is an efficient method to improve the overall performance.

The rest of this paper is organized as follows. In Section 2, we review the Sobel-Wald and Whitehead-Brunier tests. The combined double sequential weighted probability ratio test (denoted by combined 2-SWPRT) is proposed and its properties are given in Section 3. Simulation results are provided in Section 4 and some conclusions are in Section 5. All technical details are given in Appendix.

2. Methodology Review

For one-sided composite hypotheses versus , the SPRT is optimal in the sense that it minimizes the expected sample sizes at and , and the 2-SPRT has (approximately) minimal maximum expected sample size over among all sequential and nonsequential tests with the same error probabilities. Given the well-known optimality properties of the SPRT and 2-SPRT, it is natural to use the SPRTs and 2-SPRTs as the component tests to construct the sequential tests for problem (2), respectively. In this section, we briefly review the Sobel-Wald and Whitehead-Brunier tests.

For testing problem (2), the generalization of errors of types I and II is expressible in terms of a 3 × 3 error matrix , where for . However, under some mild conditions, Sobel and Wald [11, pages 504-505] and Armitage [12, pages 142-143] showed that and are zero, which can be verified by the simulation results in Section 4. It becomes apparent that in the general case we have at most four “degrees of freedom” in choosing an error matrix. Without loss of generality, we consider as a sequential test for problem (2), where is the stopping rule and is the decision rule ( means accepting , ). Set , and . Given positive vectors and (, ),is the set of all sequential tests with error probabilities controlled by and .

(1) Sobel-Wald Test. Since the hypotheses , , and are ordered, the sequential testing of problem (2) can be constructed by combining the following two one-sided composite hypotheses and :Sobel and Wald [11] proposed operating and by the SPRTs simultaneously. For all , define . The stopping and decision rules of determined by the SRPT arewhere is the indicator function and and are the boundary parameters (), which are usually set as to meet requirements on the error probabilities. When and , Sobel and Wald [11] showed the event is impossible. The stopping and decision rules of the Sobel-Wald test are defined as The Sobel-Wald test is optimal in the sense that it minimizes the expected sample sizes at and among all sequential and nonsequential tests whose error probabilities satisfy . However, its expected sample sizes at other parameters over may be unsatisfactory.

(2) Whitehead-Brunier Test. In order to minimize the maximum expected sample size under constraints (3), Whitehead and Brunier [15] applied the 2-SPRT to operate and , instead of the SPRT. As in Lorden [21], let be the Kullback-Leibler (KL) information number. Define and bySet such that , , where is the cumulative distribution function of the standard normal distribution, , and , . LetThe stopping and decision rules of determined by the 2-SPRT are where and are the boundary parameters (). The conservative values of and are and , in the sense that the real error probabilities may be much smaller than and , respectively. The stopping and decision rules of the Whitehead-Brunier test are defined as

3. Optimality Criterion and Combined 2-SWPRT

For testing problem (2), if we prefer to accept and this preference is the stronger the smaller . Similarly, if we prefer to accept , and we prefer to accept if . However, we have no strong preference between and if , and we also have no strong preference between and if . In these cases, we need more observations for decision. Thus, when the error probabilities satisfy , we focus on reduction of the expected sample sizes over the indifference-zones in applications. Let be a nonnegative weight function which is sectionally continuous on and , respectively, and satisfies . We define the weighted expected sample size as to evaluate the overall performance of sequential test plans on . The choice of should be chosen according to practical needs (Sobel and Wald [11]). For example, let be uniform weights when there are no differences on ; let be assigned more weights when we focus more on reducing the expected sample size on these parameter points. As an overall evaluation, the integrates the performances on by weighting the expected sample sizes.

Motivated by Wang et al. [20], we propose operating and by the 2-SWPRT. Specifically, the stopping and decision rules of by the 2-SWPRT are wherewhere and are the boundary parameters (). Hence, the stopping and decision rules of the combined 2-SWPRT are defined as Some features of the combined 2-SWPRT are provided in the following theorems, whose proofs are provided in appendices.

First, we show the error probabilities of the combined 2-SWPRT can be easily controlled and the stopping time is finite.

4. Simulation Studies

In this section, we conduct simulation studies to examine the performances of the combined 2-SWPRT, the Sobel-Wald test, and Whitehead-Brunier test based on the normal and Bernoulli distributions. In particular, we considered two weight functions for as follows; uniform weights: ; KL weights: , where . As in Wang et al. [20], the corresponding formulations of the statistics and can be obtained. The boundaries of the tests are determined through Monte Carlo trials, which make the relative differences between the real error probabilities ( and ) and the required ones ( and ) within ; that is, and .

Given the boundaries, we obtained the simulated to approximate integral (12) as follows. Let and be discrete as the finite sets of parameters and with increase , respectively. Denote and the weight function is calculated based on ; that is, for KL weights. We also compute the RMI to assess the relative efficiency between different test plans. According to Wang et al. [20], we define where is the smallest among the compared tests, that is, the Sobel-Wald test, the Whitehead-Brunier test, and the combined 2-SWPRT. A test plan with a smaller value is considered better in its overall performance.

4.1. Test for the Normal Mean with Known Variance

Suppose are i.i.d. from , , , and . According to Lorden [21], we have and . The stopping boundaries are obtained as follows:(1)for the Sobel-Wald test, and ;(2)for the Whitehead-Brunier test, ;(3)for the combined 2-SWPRT, for the uniform weights and for the KL weights, respectively.As expected, we found that and of these three tests are equal to 0. Set . Through another simulation study with replications, the and are presented in Table 1. Similarly, the expected sample sizes for are illustrated in Figure 1.

Table 1: WESS() and RMI() for testing normal mean.

Figure 1: Expected sample sizes for testing normal mean, , , and .

It is clear that the combined 2-SWPRTs have the smallest in all cases. In fact, compared with the Sobel-Wald and Whitehead-Brunier tests, the of the combined 2-SWPRT has been reduced by and for the uniform weights, and and for the KL weights. Meanwhile, in terms of the , the combined 2-SWPRT also performs best overall.

From Figure 1, it also can be seen that the expected sample size of the combined 2-SWPRT is slightly larger than the Whitehead-Brunier test when the true parameter is close to () and almost the same as the Sobel-Wald test when the true parameter is close to or (). When the true parameter belongs to , the combined 2-SWPRT performs better than the Whitehead-Brunier test and is comparable with the Sobel-Wald test.

4.2. Test for the True Proportion of a Bernoulli Distribution

Suppose are i.i.d. random variables from the Bernoulli distribution and . The three composite hypotheses’ testing problem is where . Let , , , , and . According to (9), we havesuch that and in the Whitehead-Brunier test and combined 2-SWPRT. The stopping boundaries are obtained as follows: (1)for the Sobel-Wald test, , , , and ;(2)for the Whitehead-Brunier test, , , , and ;(3)for the combined 2-SWPRT, , , , and for the uniform weights and , , , and for the KL weights.In this case, the values of . Set . Through another simulation study with replications, the and are presented in Table 2. Similarly, the expected sample sizes for are illustrated in Figure 2.

It can be seen from Table 2 that the combined 2-SWPRT still has the smallest and for the Bernoulli distribution. Meanwhile, from Figure 2, we have similar conclusions as those in the normal distribution cases in Section 4.1.

5. Summary

In this paper, we propose the to evaluate the overall performance on the indifference-zones for three composite hypotheses’ testing problem. In order to minimize to control the expected sample sizes, we developed a new sequential test by utilizing two 2-SWPRTs simultaneously. We have shown the proposed test is an asymptotically optimal test in the sense of asymptotically minimizing the expected sample sizes on the indifferent-zones.

According to the simulation results, compared with the Sobel-Wald and Whitehead-Brunier tests, we conclude that the proposed test has the following merits: it has the smallest and ; when the true parameter is close to , the proposed test has comparable performance with Whitehead-Brunier test; when the true parameter is close to or , it has almost the same results as the Sobel-Wald test; when the true parameter does not belong to , the proposed test also performs better than the Whitehead-Brunier test and has comparable performance with the Sobel-Wald test; the proposed test is easy to implement and can be extended to multihypothesis testing problems. Future work will be concerned with the method of determining the boundaries in an analytical way instead of the Monte Carlo method.

Appendix

Proof of Theorem 1. Let , . Note that is a supermartingale under , . Therefore, for all ,On the other hand, following Lemma 1 of Chen and Hickernell [22], for any positive integer and , we have Thus, Combining (A.1) and (A.3), we haveIn particular, setting we have . Similarly, we can prove that , , and with , , and , respectively.

Proof of Theorem 2. If is a sectionally continuous function, according to Theorem 3.2 of Wang et al. [20], we know that for all and , there exists , such that when and ; for all and , there exists , such that when and , where .Noting that is convex, we have . It is easy to choose such that Let . Then, we have . Similarly, for all and , we can prove that there exist and such that . Thus, we have

Proof of Theorem 4. Using Hoeffding inequality (see Hoeffding [23]), we know so it suffices to show According to Theorem 3.3 of Wang et al. [20], for all , Since , when and , we have Therefore, for all , Similarly, for all , we have Combining two inequalities (A.12) and (A.13), we have According to (A.8) and (A.14),Similarly, we can prove that

Proof of Theorem 5. Using Lemma 3.6 of Chen [24], for all , we know Similar to Theorem 4, for all , we have For all , there isAccording to (A.18), (A.19), and Hoeffding inequality, we have Similarly, we can prove that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the Academic Editor Antonino Laudani and an anonymous referee for their insightful comments and suggestions on this paper, which have led to significant improvements. This work was supported by the Postdoctoral Science Foundation of China (2014M560317), the National Science Fund of China (11271135, 11471119, 11371142, and 11101156), the Fundamental Research Funds for the Central Universities, the 111 Project (B14019), and the Program of Shanghai Subject Chief Scientist (14XD1401600).

P. Armitage, “Sequential analysis with more than two alternative hypotheses, and its relation to discriminant function analysis,” Journal of the Royal Statistical Society. Series B. Methodological, vol. 12, pp. 137–144, 1950.View at Google Scholar · View at MathSciNet